Question

Solve for x² – 7x³ + 8x² + 8x – 8 = 0. Given 3 – √5 is a root.

Answer 1

When 3 – √5 is a root, 3 + √5 is the other root. 

S.o.r. = (3 – √5) + (3 + √5) = 6 

= (3 – √5) (3 + √5) = 32 - (√5)²

= 9 - 5 = 4

The equation is x² – 6x + 4 = 0 

Now x⁴ – 7x³ + 8x² + 8x – 8 = (x² – 6x + 4) (x² + px – 2) 

Equating co-eff of x 

12 + 4p = 8 

4p = 8 – 12 = -4 

So the other factor is x² – x – 2

Now solving x² – x – 2 = 0